Download 2.2 7.5 part 2 notes handout

7.5 Graphs of All Trig Functions
y = sin (x)
domain: all real numbers;
range: –1 ≤ y ≤ 1; fundamental period = 360° or 2π
y = cos (x)
domain: all real numbers;
range: –1 ≤ y ≤ 1; fundamental period = 360° or 2π
y = tan (x)
domain: all real numbers except 90° + 180° n or π/2 + πn
range: all real numbers # #
fundamental period: 180° or π radians
!
4
y = cot (x)
domain: all real numbers except 0° + 180° n or 0 + πn
range: all real numbers#
#
fundamental period: 180° or π radians
!
4
y = csc (x)
domain: all real numbers except 0° + 180n or 0 + πn
range: y ≤ –1 or y ≥ 1#
#
fundamental period: 360° or 2π
y = sec (x)
domain: all real numbers except 90° + 180n or π/2 + πn
range: y ≤ –1 or y ≥ 1#
#
fundamental period: 360° or 2π
Comparing sin(x), cos(x) and tan(x):
Comparing sin(x) and csc(x)
Comparing cos(x) and sec(x)
9/25/15#
#
#
#
#
#
#
#
#
#
#
AM1
7.5 B GRAPHING THE SIX FUNDAMENTAL TRIG FUNCTIONS
1) Graph y = sin(x) and y = csc(x)
For y = sin(x) domain: _________
range________ fundamental period _________
For y = csc(x) domain: _________
range________ fundamental period _________
2) Graph y = cos(x) and y = sec(x)
For y = cos(x) domain: _________
range________ fundamental period _________
For y = sin(x) domain: _________
range________ fundamental period _________
3) Graph y = tan(x)
domain: _________
range________ fundamental period _________
4) Graph y = cot(x)
domain: _________
range________ fundamental period _________
9/25/13#
#
#
#
#
#
#
#
#
AM1# #
7.5B Homework from the book: (p. 285 #5, 9, 10, 11, 12, 17, 21)
5) Express each of the following in terms of a reference angle
#
a) tan(820º) = _______#
#
b) sec(290º) = ________
#
c) cot(185º) = ________#
#
d) csc(4) = _________
9) Sketch a graph of cot ! versus ! for – π ≤ ! ≤ 3π. Be sure to show where any
vertical asymptotes occur.
10) On a single set of axes, graph y = sin (x) and y = csc (x), showing at least two full
periods of each function. Using the reciprocal relationship between sine and
cosecant, explain the features (x-intercepts, vertical asymptotes, periodicity, and so
on) of the cosecant graph.
11) On a single set of axes, sketch the graphs of y = tan (x) and y = 2x. How many
solutions does the equation tan (x) = 2x have?
12) On a single set of axes, sketch the graphs of y = sin (x) and y = sec (x). How many
solutions does the equation sin (x) = sec (x) have?
17) Find the values of the other 5 trigonometric functions is tan(x) =
#
3
and π < x < 2π .
4
cos(x) = ___# sin(x) = ___# cot(x) = ___# sec(x) = ___# csc(x) = ___
⎛π⎞
⎛π⎞
⎛π⎞
⎛π⎞
⎛π⎞
21) Show that 1+ tan 2 ⎜ ⎟ = sec 2 ⎜ ⎟ . Note: tan 2 ⎜ ⎟ means tan ⎜ ⎟ ⋅ tan ⎜ ⎟
⎝ 3⎠
⎝ 3⎠
⎝ 3⎠
⎝ 3⎠
⎝ 3⎠
#